REPORT NO. 1492

STABILIZATION OF A LIQUID-FILLED SHELL BY INSERTING A

CYLiNDRiCAL PARTiTiON IN THE LIQUID CAVITY

by

j. T. Fras ierW. P. D'Amico

August 1970

Approved for public release; distribution unlimited.

u.s. ARMY ABERDEEN RESEARCH AND DEVELOPMENT CENTERBALLISTIC RESEARCH LABORAT{iRIESABERDEEN PROVING GROUND, MARYLAND

Destroy this report when it is no longer needed.Do not return it to the originator.

The findings in this report are not to be construed asan official Department of the Army position, unlessso designated by other authorized documents.

B ALL 1ST I ~ RES EAR C H LAB 0 RAT 0 R I E S

REPORT NO. 1492

AUGUST 1970

STABILIZATION OF A LIQUID-FILLED SHELL BY INSERTING A CYLINDRICAL PARTITIONIN THE LIQUID CAVITY

J. T. Frasier

Terminal Ballistics Laboratory

w. P. D'funico

Exterior Ballistics Laboratory

Presented at and published in the proceedings of the 8th Navy Symposiumon Aeroballistics

Approved for public release, distribution unl1lrited.

RDT&E IW542703D344

ABE R DEE N PRO V I N G G R 0 U N D, MAR Y LAN D

B ALL 1ST I eRE SEA R C H LAB 0 RAT 0 R I E S

REPORT NO. 1492

JTFrasier/WPD'Amico/lseAberdeen Proving Ground, Md.August 1970

LIQUID~F!LLED SHELL BY INSERTING A CYLINDRICAL PARTITIONIN THE LIQUID CAVITY

ABSTRACT

In July 1968, the XM613, a 107mm, WP mortar shell, was range tested

at Yuma Proving Ground. The projectile had a history of flight instabilities

when the WP was in the liquid state. Specially modified rounds incor­

porating a cylindrical partition mounted concentrically to the central

burster experienced stable flights. The rationale of inserting cylindrical

partitions in liquid... filled shell is explained as being a straightforward

and flexible method of changing internal geometries to produce stable flight

by means of understood design procedures.

3

TABLE OF CONTENTS

Page

ABSTRACT . . . . . . . .

LIST OF ILLUSTRATIONS.

I. INTRODUCTION . . . . .

II. LIQUID-FILLED PROJECTILE STABILITY

Stewartson's T'neory of Stabili ty .

Assumptions of Stewartson's Analysis

Results of Stewartson's Analysis ..

3

7

9

9

.,,,• • • • • lU

11

13

III.. STABILITY OF A SHELL WITH A CENTRAL ROD ALONG ITSPAYLOAD COMPAR1MENT. . . . . . . . . . . . . . . . . . 20

IV. RATIONALE FOR THE INSERTION OF A CYLINDRICALPARTITION.

V. CONCLUSIONS. .

REFERENCES . .

DISTRIBUTION LIST.

5

26

27

29

LIST OF ILLUSTRATIONS

Figure No. Page

1

2

3

4

Resonance Curve for Liquid-Filled Projectile .

Typical Shell Cavities ...

Physical Characteristics of the XM613 .

Nondimensiona1 Frequencies Versus Partition OutsideRadii . . . . . . . . . . . . . . . . . , . . . ·

7

19

22

23

25

I. INTRODUCTION

Over the past decade investigators at the U.S. Army BallisticsResearch Laboratories have conducted theoretical and experimental studiesof the causes of flight instability of liquid-filled projectiles.Recently, these studies led to a rationale for the stabilization of suchprojectiles by incorporating withi~ their payload compartment, an axiallyaligned cylindrical partition [la] A limited, but successful, fieldtest of the design rationale was made in July 1968.

The XM 613, a l07mm, spin stabilized, WP (white phosphorous)mortar shell, was range tested at Yuma Proving Ground. The projectilehad a history of flight instabilities when the WP was in the liquid state.Specially modified rounds incorporating cylindrical partitions experiencedstable flights. Limited resources allowed the testing of only two rounds,and therefore conclusive proof testing of the rationale cannot be claimed.However, the insertion of cylindrical partitions in liquid-filled projectilesis a straight-forward and flexible method to produce stable flight bymeans of well-founded design procedures. In fact it is the most flexiblemeans for "a priori" design of a stable liquid~filled-projectileavailableto munitions designers today. This method is not seen as a panacea, butit does represent a major improvement over previous "ad hoc" procedures.

Our intention in the present paper is to explain the rationalefer using a cylindrical partition to achieve well-behaved flights ofliquid-filled shell. This is a simple task if we assume our audience isfamiliar with the work done at the BRL over the past ten years. Thisassumption is not likely to be valid, however. We will begin ourpresentation with a brief discussion of the problem of liquid-filledprojectiles and our current understanding of the causes of their ills.Having done this, the use of cylindrical partitions in the payloadcompartment to achieve stability will be explained.

II. LIQUID-FILLED PROJECTILE STABILITY

WP rounds, one of the most cowmen types of liquid-filledprojectiles, have long been infamous for their poor flight behavior,(WP melts at ll2op). A major step toward understanding the physicalrealities of this behavior was achieved by Stewartson when he published

Numbers in square broakets denote referenaes found at the end of thepaper.

9

an analysis of the stability of a spinning top containing liquid in acylindrical cavity [2]. This work provided a clear definition of thebasic mechanisms causing flight instabilities of liquid-filled projectilesand was the basis for further analytical and experimental research. Asummary description of the current level of knowledge concerning liquid~

filled projectiles is given very conveniently by using the conditions,assumptions, and results of Stewartson's analysis as a foundation.Theoretical and practical advances are, in large measure, the consequenceof relaxation of the assumptions of Stewartson's analysis.

STEWARTSON'S THEORY OF STABILITY

Stewartson's theory concerns the flight stability of a spinningshell with a right circular cylindrical cavity either wholly or partiallyfilled with liquid. Results of the theory show that growth of thenutational component of the projectile's yaw is possible under adversecombinationsof the geometrical and physical characteristics of theprojectile and its liquid filler. Instabilities are a consequence ofthe nutational frequency of the shell being hazardously close tocertain natural frequencies of the liquid. This can be described as acondition of resonance. When this condition occurs, oscillations of theliquid produce a periodic moment (couple) on the shell casing and leadto a growth in yaw.

The theory is valuable for several reasons. First, it providesa clear understanding of the physical phenomena through which liquidsproduce instabilities in spinning projectiles, namely, the resonance typebehavior mentioned above. This basic mechanism of instability apnliesto cavities of all geometries and not just cylinders. Second, theassumptions and conditions of the theory supply a useful framework todiscuss the work at the BRL. The advances represented by the latterefforts are, in many instances, the consequence of modifications orrelaxation of Stewartson's assumptions and conditions.

Stewartson's theory is based upon assumptions and stipulationsthat define the situations for which it is valid. Here we will reviewthe most important of these factors and attempt to point out theirphysical significance. When appropriate, research advances by otherinvestigators will be mentioned and references to their work cited.Certain of the stipulations are absolute requirements in that if theyare not satisfied the theory is invalid. Others are taken as a matterof convenience to simplify the analysis and to clarify the role of theliquid in causing flight instability. An example of an assumption ofthe latter type is that the overturning moment is the only significantaerodynamic force or moment acting on the shell. Drag, etc., can beincluded in the analysis but are not essential to its development. there­fore, we shall neglect them to maintain focus on the basic features ofinteractions between the shell and its liquid. Distinctions betweenthe two tynes of assumptions will become clear in the course of discussion.

10

Assumptions of Stewartson's Analysis

The Cavity in the Shell is a Right Circular Cylinder wnose Axisis Parallel to the Spin Axis of the Projectile. Immediately

we see that the theory is restricted to shell with cylindrical cavities.Hence, direct quantitative application of its results is limited to asingle geometrical shape. Wedemeyer, however, has achieved a modificationof Stewartson's theory through which it is possible to design for othershapes; specifically, cavities whose radii change slowly along theirlength [3a]. As a consequence, we are able to perform analyses onmany practical cavity goemetries. To use Wedemeyer's modification,one must have a working knowledge of the basic Stewartson analysis.

The Aerodynamic Overturning Moment is the Only External Forceor Moment Affecting the Flight of the Shell. We explained

above that this assumption is taken for convenience. Gravity andother aerodynamic effects can be taken into account through conventionalprocedures but are not essential to the theory.

The Shell Flies wit~_Constant Translational Velocity a~~i~.

Tnis assumption is consistent but implies' that the liquid doesnot influence the spin rate of the projectile. In practice these conditionsare never satisfied. For a period after the projectile leaves the gun,the shell's spin decreases because it must spin-up the liquid. Subsequentto liquid spin-up, the shell experiences spin decrease due to drag. ­Furthermore, the projectile encounters translational drag. In generalhowever, once the transient phase of liquid spin-up complete, the drageffects are sufficiently small for the current assumption to be reasonablefor most projectiles.

Finally, we remark that this assumption deals with the shellcasing - not with the liquid - and relates to the equation of motionwritten for the projectile. Assumptions below concern the motion ofthe liquid.

The Gross Motion of the Liquid is a Rigid Body Translation andSpin Identical to the Translation and Spin of the Projectile.

This assumption, in conjunction with the one above, restricts ourconsiderations to the full spin condition of the liquid. The theorydoes not consider situations where the shell casing and liquid haveunequal rigid body spins, nor does it take account of variations inthe spin of the liquid.

Wedemeyer [3b] and Scott [4] have performed analyses that allowus to calculate the time required for the liquid to spin-up after aprojectile leaves the muzzle. Hence, in practical situations, we candetermine when Stewartson's full-spin assumption becomes valid. Usually,it is soon after exit from the muzzle. Furthermore, a semi-empiricalfulalysis is available to determine whether instability is likely tooccur during the transient spin-up process [5].

11

The Spin of the Liquid and the Dimensions of the CylindricalCavity Satisfy the Condition.

where a

2 2a ~ »qc

cavity radius, inches,

(1)

~ spin rate of the projectile (and therefore the liquid),rad/sec,

q

2c

= d f h d · d d . / 2magnitu e 0 t e resolve gravlty an rag vectors, ln sec,

height of the cavity, inches.

The physical significance of this assumption is that centrifugal forcesexerted on the liquid due to its spin far outshadow any forces imposedby gravity or drag. A consequence of the assumption is that the liquid(except when the cavity is completely filled) has the shape of acylinder with a hollow, cylindrical core*.

Equation (1) must be satisfied for Stewartson's theory to beused. If a shell experiences high drag along with a low spin ratethere is a possibility the relation will be violated. Then the liquidwill not have a cylindrical core, but will develon a paraboloidal surface.Designers should always verify Equation (1) is satisfied to avoidimprudent application of the theory.

It should be understood that the current assumption is not theexternal forces assumntion. The former condition concerns externalforces ~ld moments acting on the shell casing and their effect onthe motion of the shell. The present assumption concerns the effect ofgravity and d~ag on the behavior of the liquid.

The Mass of the Liquid is Small Compared to the Total Mass ofthe Shell. This a;sumption is one of convenience. It is

satisfied for many shell and simplifies the equations of motion for theliquid-shell system. We use it here for these reasons.

The Liquid is Incompressible and Inviscid. The assumption ofincompressibility is reasonable for the liquids encountered in actualprojectiles. Viscous effects, however, can influence the behavior ofliquid-filled projectiles. Fortunately, Wedemeyer has provided ananalysis to account for these effects [3c]. His analysis involves aboundary layer correction to the basic, inviscid theory of Stewartson.

The Final Assumptions Concern the Nature of any Variations tothe Rigid Body T~anslation and Spin of the Shell and Liquid.Any Disturbance to the Shell's Motion is Restricted to SmallAmplitude Perturbations Superposed on its Gross Translationand Spin. Correspondingly, the Liquid is Assumed to Experienceonly Small Amplitude Perturbations to its Large Scale Translationand Spin. The assumption about the shell is the familiar small

yaw situation associated with the linearized equations of yawing motion.

*Actually a paraboloid whose vertex is far from the shell.

12

Similarly, the assumption imposed on the liquid linearizes the equationsdescribing its behavior. By virtue of linearization, the equations forthe liquid motions can be solved and their result incorporated into theequations of the motion for the shell.

Results of Stewartson's Analysis

All the basic assumptions and conditions underlying Stewartson'stheory were presented above. From these assumptions, we can make aqualitative statement of our problem. Namely, determine the conditionsfor which a symmetric, rapidly spinning projectile will experience aflight instability as a consequence of having liquid (at full spin) ina cylindrical, axially symmetric cavity. Stewartson attacked thisproblem in two phases, and it seems most effective to describe hisanalysis in a similar fashion. First, he considered the behavior ofthe liquid in a state of rapid rotation within a container that couldperform motions similar to those of the yawing motion of a shell. Hethen combined the problem solution he found for the liquid with theequations of motion for a shell. Upon analysis of the resulting equationsit was. found that under certain adverse conditions the yaw of the shellwill grow without limit.

To describe the behavior of the liquid, we remember that it isconfined in a container and that its basic motioll involves rigid bodyspin about an axis with fixed direction. Upon assuming that the axisof the container is subjected to a small disturbance similar to theyawing motion of a shell, it is necessary that the liquid also experiencesa disturbance to its basic motion because it must follow the walls ofthe cavity. Stewartson's solution shows that the liquid conforms tothe cavity motion through the exitation of small amplitude oscillationssuperposed on the rigid body motion. There is an infinite number ofdiscrete frequencies for these oscillations - the natural frequencies(or eigenfrequencies) of the spinning liquid. For an arbitrary motion ofthe container all the natural frequencies will be excited, but in varyingdegrees. If, however, the container performs a yawing motion at certain ofthe eigenfrequencies of the liquid, oscillations at this frequency becomepredominant, that is, a condition of resonance is established. As we shalldescribe later, it is this resonance that leads to the instability of ali1uid filled projectile.

We should emphasize that the oscillations performed by theliquid are of small amplitude. Sloshing does not occur, but a wavepattern is established in the longitudinal, radial, and circumferentialdirections of the cavity and there are mode numbers* associated witheach direction. For problems of projectile stability, an infinity ofthe possible longitudinal and radial modes are significant theoretically,but only the first circumferential mode is important. This circumstanceis a result of the fact that the pressure fluctuations produced in the

* These can be thought of as fundamental Wave patte~s and harmonics.

13

liquid by this mode lead to a periodic couple (with the same frequencyas that of the oscillating liquid) on the walls of the container. Itis this couple which renders a projectile unstable.

(2)

Now, we must explain how the natural frequencies of the liquidin aare determined. Stewartson 's theory gives these frequencies

complicated relation of the form

b~lw r.2U. f c 1, ") 3,T nnj r2 n t C2j + 1) a 2J7 ~;

j 0, 1, 2,

where n

w .nJ

T .nJ

2a

radial mode number (the number of nodes in the radial wavepattern),

longitudinal wave number (2j + 1 = number of nodes in thelongitudinal wave pattern),

natural frequency of the njth node,

.ththe non-dimensional eigenfrequency of the nJ mode,

diameter of the cavity,

2b = diameter of the cylindrical air core,

2c cavity length.

Several aspects of Equation (1) should be noted. First, theeigenfrequencies of the liqu~d ~re dependent upon the cavity geometrythrough the ratios cia and b /a. The ratio bLla2 is the air volume inthe cavity expressed as a fraction of the total cavity volume. Hence(1 b 2/a2) is the fraction of the cavity occupied by liquid. Next,we note that the eigenfrequencies depend upon the longitudinal modenumber through the ratio c/a(2j + 1) appearing as a variable in f n . Thisis a fortunate circumstance, because once Tn; is known for a set offixed values of c/a(2j + 1), b2/a2 , and n, the eigenfrequencies areknown for all longitudinal modes for which c/a(2j + 1) equals the setvalue. Finally, Equation (2) shows the frequencies are linearly relatedto D, that is, Tnj is independent of D.

As mentioned above, the function of f n in Equation (1) is acomplicated one, and it must be evaluated numerically through machinedetermination of the poles (singularities) of another equation appearingin the Stewartson analysis. This has been done and the results, theliquid eigenfrequencies, tabulated so that it is possible to makequantitive use of Stewartson's and subsequent work [lb].

This completes our discussion of the natural frequencies ofthe spinning liquid, which are dependant upon the cavity geometry,i.e., (c/a and b/a). Now, we turn to the questions of how and when

14

an instability of a liquid-filled shell is produced by the oscillatingliquid. To begin we recall our assumptions that the overturning momentis the only significant aerodynamic force or moment acting on theshell and that we are dealing with small yaws. Under these conditi~~s,

the motion of the shell (without the liquid) is governed by the relations

A A e irtTt(3)

0

I2

I T2

I T +X

0 (4 )Y x 4TS

y

where I and 1y are, respectively, the axial and transverse moments ofinertiaXof the shell, and

A is the complex yaw,

t is time,

1 is the non-dimensional frequency of the motion of the shell,

s is the gyroscopic stability factor.

(5)

Since Equationthe she 11 and

Nutational frequencyTn

Equation (3) represents the form of the motion ofvalues of T are provided by solution of Equation (4).is quadratic, the latter are found easily:

1 Ix2" I (1 + 0)

Y

the(4 )

1 IxT = 2" I (1 - o~P y

Precessional frequency (6)

s

where a = ~ 1

- 2\)

4M

1s

-\)

M

Pa air density

S reference area of shell, usually the cross-sectional area,

n = twist of rifling, calibers per turn,

15

d diameter of the shell

m = mass of the shell

eM1'1

a.

md2

/1y

aerodynamic overturning moment coefficient

It is advantageous to recall here that the shell is stable (theyaw does not grow with time) so long as Tn and Tp are real quantities.To achieve this situation we must have s greater than one, a familiarcondition for the gyroscopic stability of a projectile. If s is lessthan one, a and therefore Tn and Tp become imaginary, and an exponentialgrowth of yaw occurs.

Earlier, we stated that the oscillating liquid produces a momenton the casing of the projectile. Now we must describe that moment infunctional form and modify Equation (4) to inClude its effect. Stewartsonshowed that when the frequency, T, of the projectile is near anyone ofthe natural frequencies, Tn;' of the liquid, the moment anplied to theshell casing is given by -

6 2pa [2R(T ,)] /4c

nJ

T - T ,nJ

(7)

where ML

= the moment exerted on the shell by the liquid,

T the frequency of motion of the shell,

o = the density of the liquid,

ReT.) a (small; constant depending upon T . and is always positive,nJ "the residue". nJ

The quantity ~; is available in the tabulation of liquid eigenfrequenciesreferred to above [Th]. We see from Equation (7) that it governs themagnitude of the liquid-moment for a given cavity and frequency, Tn"Each possible frequency and modal configuration of the liquid invol~esa specific value of Rnj . With regard to the residue, it is convenientto point out a significant feature of its behavior. Namely, for anyspecific value of frequency, Tnj' the residue decreases greatly foreach successively larger value of n (i.e., Rnj decreases with increasingradial mode number). Thus, the higher radial-modes produce relativelyweak liquid-moments. In practice it has been found that modes beyondn = 2 are seldom strong enough to cause projectiles to be unstable.

16

The moment due to the liquid is a forcing function on the motionof the shell. Thus to account for the presence of the liquid in theequation of motion of the shell we add Equation (7) to the right-hand-sideof Equation (4) and obtain

2I '[ - I TY X T - T

o(8)

where for convenience, TO has been written in place of Tn· to emphasizethat we are now thinking of a specific fluid frequency. ay solvingEquation (8) for T, the frequency of the sheilis motion and the conditionsunder which the liquid can produce an unstable flight are determined,(that is, the conditions for which T has an imaginary part assuming, ofcourse, that s > 1). Here, we shall only summarize the results of thissolution. It is found that when Tn is close to Tn, the precessionalfrequency, no instability occurs. ~However, if Tor-is near the nutationalfrequency, Tn' Equation (8) has the roots

T

6 2pa [2R(T,J]

u

4cI ax

(9)

Ao

exp

= A expa

ClO)

(9) .imaginary

exp (astt) ,

instability is provided immediately by Equationunder the radical is negative, T has a negativesubstituting Equation (9) into Equation (3):

W _HTn T O)2" 2 I

The condition forWhen the quantitypart as we see by

where

Thus, an exponential growth of the nutational component of yaw occurswhen

< 0

2- T \0)2 I,IT( n

17

or, written more conveniently, when

where 5, "Stewartson's Parameter", is

5 25 = pa (2R)

I aCc/a)x

The condition forinstability

(11)

When Equation (11) is satisfied, the rate of growth of yaw is [Equation (10)].

or

a .!.. ~ _ 1)22\/U- \.1 0 n (12)

2

.JSa =

Figure I is a plot of (2/5 1/ 2)a against (1 0 - 1 n)/51/ 2 . This

curve, as well as examination of Equations (11) and (12), shows the yawgrowth rate to be largest when 10 = 1n . For non-zero values of 1 - 1n,the yaw growth rate decreases until it vanishes for ITo - Tnl= Sl~2.

Equations (11) and (12) are the basic results of the 5tewartsontheory. They permit us to calculate the conditions producing instabilityin a given projectile (and therefore a means to avoid these instabilities)and to calculate the strength of the instability, provided, of course,that the assumptions of 5tewartson's analysis are satisfied. Moreaccurately, we should say "provided that these conditions are satisfiedor that advantage is made of the advances of Karpov [5], Wedemeyer [3]and Scott [4] to relax these restrictions." The work of these investigatorswas referenced during discussion of the assumptions of Stewartson'sanalysis. This work is extremely important in that it demonstrated thestrengths and weaknesses of the original analysis and made the necessaryadvances for the practical application of the theory. Space does notpermit any discussion of the details of this work. It can be summarizedquickly, however, by pointing out that this work considers three pointsof concern that must be included in the analysis of any genuinelypractical situation. Namely,

• Viscous effects in the liquid filler [3c],

• Certain types of non-cylindrical cavities, including thosewith profiles similar to the ogival shape of conventionalartillery projectiles [3a],

• Liquid spin-up effects [3b].

18

-I

(:2 )

a -.lIS

STEWA~RTSON' S INVI:SCID/ PREDICTION

----------:.::-;1-- (To - Tn) / .IS

III. STABILITY OF A SHELL WITH A CENTRAL ROD ALONG ITS PAYLOAD COMPARTMENT

Ine preceeding discussion has centered about the problem ofthe stability of a projectile with a payload cavity that is eitherpartially or completely filled with liquid. For the original Stewartsoncase this cavity is a right circular cylinder. However, owing to theefforts of Wedemeyer this restriction can be relaxed to treat cavitieswhere a nrofile radius varies slowly with axial distance [2]. Animportant feature of these analyses, however, is that when the cavity ispartially filled, a cylindrical air core runs along the axis of thecavity. A problem very similar to this and one that is important tothe rationale for projectile design involves a core of rigid materialalong the center of the cavity rather than a flexible air core.

Let us substitute an axially rigid core for Stewartson'sflexible air core. Analytical treatment of the problem of flight stabilityof a projectile carrying liquid in a payload compartment consisting ofa right circular cylinder with an axially aligned rigid core is quitesimilar to that of the Stewartson problem. This problem has recentlybeen solved by Frasier and Scott and has results completely analyogousto those discussed earlier [Ie]. Namely, yaw of the projectile growswithout limit if the nutational frequency of the projectile is sufficientlyclose to certain of the eigenfrequencies of the spinning liquid payload.More specificially, the constraints apply to the analysis; and the resultsexpressed by Equation (1) through (12) apply with only one qualification.That qualification is that the dimension b used in those equationsshould be replaced by d, where d is the radius of the rod in the cavity.The residues for this case are numerically distinct from the air coresolution.

We should emphasize that the analysis involving the rod requiresthat all the volume of the cavity after the rod is inserted is completelyfilled with liquid. Hence, the side walls, end faces, and rod are wettedby liquid at all times. We will know explain hawaII of the preceedingmethods can be combined into procedures for the insertion of a cylindricalpartition.

IV. RATIONALE FOR THE INSERTION OF A CYLINDRICAL PARTITION

The rationale for inserting cylindrical partitions in liquid­filled shell is an integrated use of Stewartson's theory and the subsequentextensions to techniques that account for non-cylindrical cavities, rigidcentral cores, and liquid spin-up. Once castings· for a liquid-carryingprojectile are made, it is not an easy task to modify the round for theelimination of shell-liquid resonances. Several "ad hoc" approaches forstabilization, longitudinal baffles being the most common, are frequentlynot successful. It is desirable to establish techniques that make possible"a priori" design control. A concept satisfying this criterion involvesthe insertion of a cylindrical partition in the payload cavity.

20

Flight instabilities due to liquid payloads are a consequenceof resonance between liquid eigenfrequencies and the shell nutationalfrequency. This matching can occur during liquid spin-up or at a fullspin condition. To alieviate such a resonant condition, the shellgeometry, the percent of fill, or the shell nutational frequency mustbe changed. Normally, the liquid payload cannot be significantly altered.A redistribution of mass could shift the shell nutational frequency,but this results in a complete aerodynamic redesign. It appears thata variation in cavity geometry is the best route. TIlis geometrymodification should not be a haphazard one.

An intelligent change in geometry can be made by using a cylindri­cal partition. To illustrate this method, consider the two cornmonshell cavities shown in Figure 2. Non-cylindrical cavities are shownwith a burster located in the nose and with a central burster. Assumethat the cavity is 9S percent filled with liquid. A cylindricalpartition could be mounted about the longitudinal center line, as indi­cated by the broken lines. These cylinders must be fastened in such away as to allow the liquid to move over the cylinder edges when it isthrown outward by centrifugal forces. Since the spinning liquid willseek the outermost position, the outer cavity will be 100 percent filledafter the liquid spin-up process is completed.

Effectively two cavities result from the insertion of a cylinderin a shell cavity. For Figure 2a, the outer cavity takes up boundaryconditions of a non-cylindrical wall and a rigid core (100 percent full).The inner cavity is the case treated by Stewartson, consisting of apartially filled cylindrical cavity [2]. A similar double cavity resultsin Figure 2~ except that the central burster may act as a rigid core [Ic]or as a partially wetted rod [6]. If the percent fill is not highenough to cause liquid-burster interference, then both Figure 2a and 2bhave the same boundary conditions. Usually a cylinder radius canbe selected that will result in a stable flight.

Assume that an analysis of some non-cylindrical cavity revealsa resonant condition. (How could a cylindrical partition be selectedto stabilize the projectile?) In the case of the XM613, a I07mrn, WPmortar shell, an unstable flight was caused by a transient resonance.The ~~613 georr~try is shown in Fi~ure 3, with a cylindrical partitionin place. It is possible to generate a table of nutational frequenciesand rigid core radii that produce resonant cnnditions for the outercavity ~ k]. This is done for rigid core, fUll spin modes. Consideringonly the first radial mode, Table I lists the longitudinal modes(j' = 2, 3, 4, and 5) for a fill ratio of 100 percent over a range offrequencies close to the ~~613 nutational frequency of 0.067.

21

tvtv

----~Figure 2. Typicial Shell Cavities

9.75

J_

All dimensions in inches

'. I 1 2 :~ I ~....".•--' L•.!, .....Lldd.L..L.l~_.j,L. 1.,,1., I I I I ,\ 1.1,1I!,u1,j,. ,,, ••

l r·2501.5001- ~ 2 SlotsLQ( ot180°

~-Detail A_I

Section B-B

PHYSICALS FOR XM613

Weight - 25.5Ibs.Axial moment ot inertia 64.0 Ibs. in 2

Transverse momen1 of inertia 740.0 Ibs. ir..2

Gyroscopic staoility factor 1.43Spin rate 68 radians / secondFi J; ratio 9570Nutational frequency 0.067Specific gravity 1.72Kinematic viscosity 1.2 centistokes

Figure 3. Physical Characteristics of the XM6l3

23

(U) TABLE I

d. 2 d. 3 d. 4 d. 5J= J= J= .1=

1 inches inches inches inches

0.00 0.568 0.891 1.036 1.138

0.02 0.616 0.910 1. 052 1. 156

0.04 0.653 0.927 1.066 1.166

0.06 0.686 0.944 1. 081 1.179

0.08 0.718 0.961 1.096 1. 194

0.10 0.745 0.978 1. III 1.208

A graphical representation of Table I is shown in Figure 4.The selection of a nonresonant cylinder radius, d, can easily be made.Logical choices would be 0.85" or 1.025". If values of 0.70" or 0.95"were selected, a resonant condition is produced. Once d and a cylinderwall thickness are chosen then the geometry of the inner cavity is fixed.For the XM6l3, a cylinder outside radius of L025" and a wall thicknessof 0.10" were selected. The slenderness ratio (height/diameter) of theinner cavity for the j = 0 mode was 5.27. The resulting percent offill for the inner cavity was 62.5 percent.

The new design was checked for viscous effects and transientresonances [3,7]. Conditions for the XM6l3 were such that no viscouscorrections needed to be made. Transient resonances can occur ineither of the cavities. Transient eigenfrequencies cannot be explicitlycalculated, but the yaw growth while passing through a transient instabilitycan be approximated by the following equation [5,7].

SdT /dt

o

where exo

yaw angle before transient resonance,

exl = yaw angle after transient resonance,

rt = shell spin rate,

S Stewartson's parameter,

1 = time dependent, non-dimens i onal eigenfrequency,0

t = time.

24

1.6

j =lon~Jitudinal modenunlber

1.41.21.0

. . .j:=3 J=4 J=5

I

(l8O.E)o.·~0 ").ll:.

0 ..- ......- ...----...o

0.10 -

0.06 -

0.08 -

0.02 .

0.04 .

~'I

)-.

Ul2~

LLJI::::JIOrWIa:L&..

..J<tZ0-~t-::>Z

tv..J

V1 <tZ0(J)

ZUJ~-0I

Z0Z

RIGID CORE RADIUS, d - in.Figure 4. Nondimensional Frequencies Versus Partition

Outs 'i de Radi i

If dTo/dt is computed correctly, viscous effects in the spin-up processare taken into account. Assumptions made to extend the above equationto partially filled cylinders or to non-cylindrical, solid core cavitiesare not overly restrictive. For the inner cavity of the XM6l3, a transientresonance was located for an effective percent fill of 0.597. Sinceal/a

owas calculated to be 1.35, no serious problems were expected.

It is quite nossible the cylinder radii available from plotssuch as Figure 4 could still yield serious resonances for either spin-upor full spin conditions. It might then be possible to use more thanone cylinder, especially in some of the larger rounds. A multi-cylinderdesign should still allow for fluid movement, while not decreasing thepayload capacity or impairing the manufacturing process. This was notneeded for the XM6l3, which was f~ just as shown in Figure 3.

Firings held in July 1968 at Yuma Proving Ground were part of aproject termination that included the expenditure of available hardware.Since the XM6l3 had a history of flight instabilities, blamed uponshell-liquid resonances, this afforded the design engineers a testvehicle for "ad hoc" stabilization methods. At the invitation ofpersonnel from the Artillery and Mortar Section of Picatinny Arsenaland the Weapons Development and Engineering Laboratory of EdgewoodArsenal, the BRL suggested the use of a cylindrical partition. Othermodifications made by the MUCOM agencies included the insertion ofseveral types of longitudinal baffles, the use of a gel filler repre­senting thickened WP, and the addition of a sponge material into theshell cavity in an attempt to suspend the WP, i.e., reduce fluid move­ment. Firings were made from a single tube for a single charge andelevation. Thirty projectiles were thermally cured to a surfacetemperature of l45 0 F for a period of twenty~four hours. Four to fiverounds were removed from the conditioning oven and fired within thirtyminutes until all ammunition was expended. This insured that thefiller was in a liqUid state. Of the thirty rounds fired only the tworounds that employed cylindrical partitions flew well. These two roundsfell ten meters apart at 6,000 meters.

V. CONCLUSIONS

The small number of rounds that have actually been field tested doesnot renresent statistical proof or conclusive verification of the overallability of a cylindrical partition to stabilize liquid-filled shell.It does, however, produce a high level of confidence in the technique andin the combined use of many theoretical methods. The use of a cylindricalpartition is the only available procedure that has a well-understoodtheoretical foundation for "ad hoc" stabilization.

26

REFERENCES

lao J. T. Frasier and W. P. D'Amico, "Stabilization of a Liquid-FilledShell by Inserting a Cylindrical Partition in the Liquid Cavity,"Ballistic Research Laboratories Technical Note 1713, March 1969.

lb. J. T. Frasier, "Dynamics of Liquid Filled Shell: Aids for Designers,"Ballistic Research Laboratories Report 1892, December 1967.

Ie. J. T .Frasier and W. E. Scott, "Dynamics of Liquid-Filled Shell:r"l;T1r1'l";r!:ll r!:l";?-" w;?-n !:I r&>T1?-'l"':31 Onrl " R!:Ill;c?-;r O&>c&>!:I,..rn T!:Ihn"'!:I?-n,..;&>c~J ..... -I-.I.a.,..,. .... -'-'"""'-4...L ""'u.v~'"'] ..... ~ .... .I ... u. "-'''''.11''''''.La~ .l'\.'\Ju., IJ~..L.L..L..::J,""..L,"", .I,'-"..::J"""u..L'-J. ... UUOUV.LUO""V.L-L""'.,;;;)

Report 1391, February 1968.

2. K. Stewartson, "On the Stability of a Spinning Top ContainingLiquid," J. of Fluid Mech., Vol. 5, part 4, 1959, pp. 577-592.

3a. E. H. Wedemeyer, "Dynamics of Liquid-Filled Shell: Non-CylindricalCavity," Ballistic Research Laboratories Report 1326, August 1966.

3b. E. H. Wedemeyer, "The Unsteady Flow Within a Spinning Cylinder,"Ballistic Research Laboratories Report 1225, October 1965.

3c. E. H. Wedemeyer, "Viscous Corrections to Stewartson's StabilityCriterion," Ballistic Research Laboratories Report 1325, June 1966.

4. W. E. Scott, "A Theoretical Analysis of the Axial Spin Decay of aSpin-Stabilized Liquid-Filled Shell, " Ballistic Research LaboratoriesReport 1170, August 1962.

5. B. G. Karpov, "Dynamics of Liquid-Fi lIed She 11: Instabi Ii ty DuringSpin-Up," Ballistic Research Laboratories Memorandum Report 1692,January 1965.

6. W. P. D'Amico, "Dynamics of a Liquid-Filled Shell: Rod-Liquie Inter­ference," Ballistic Research Laboratories Report 1985, June 1969.

7. Liquid-Filled Projectile Design, Engineering Design Handbook of theArmy Materiel Command, AMCP 706-165, April 1969.

27

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UNCLASSIFIEDSecuritv Classification

DOCUMENT CONTROL DATA· R&D

1. ORIGINATING ACTIVITY (Corporet. euthor) a.. REPORT S.CURITY CLASSIF'ICATIONUSAARDC Ballistic Research LaboratoriesAberdeen Proving Ground, Md.

3. REPORT TI TL E

UNCLASSIFIEDabo GROUP

STABILIZATION OF A LIQUID-FILLED SHELL BY INSERTING A CYLINDRICAL PARTITION INTHE LIQUID CAVITY

•. OESC"IPTIVE NOTES (Typ. of r.porl .nd Incluelv. al.e)

II· AU THOReS) (Flret 1M1l1I_. IIIlddl. Inltl.I. le.' neill.)

John T. Frasier, William P. DIAmico

N. ORIGINATOR". REPORT NUM'aI:ReS)

e· REPORT DATEAugust 1970

... CONTRACT OR GRANT NO.

7e. TOTAL NO. OF' PAGES32

7b. NO. OF' "I:,.s7

b. PROJECT NO. RDT&E lWS42703D344

c.

d.

Report No. 1492

8b. OTHER REPORT NOe.) (Any oth.r number. th., _y b_ •••/.,..dtltl. repor')

10. DISTRIBUTION STATEM.NT

"i

1 I. SUPPLEMENTARY NOT••

13. ABSTRACT

Approved for public release. distribution uniimited.

12. SPONSORING MILl TARY ACTIVITY

U.S. Army Materiel CommandWashington, D.C.

In July 1968, the SM613, a 107mm, WP mortar shell, was range tested at Yuma

IPrOVing Ground. The projectile had a history of flight instabilities when the WP was ~in the liquid state. Specially modified rounds incorporating a cylindrical partitionmounted concentrically to the central burster experienced stable flights. The rationalof inserting cylindrical partitions in liquid filled shell is explained as being astraightforward and flexible method of chanQinQ internal Qeometries to produce stableflight by means of understood design proced~re~. ~ L

I

UNCLASSIFIEDsecurity Cla..IRcation

, , .... 1 r"' T A. ~ C' T T"'" T r nUI~L.LA;:);:) 1 r 1 CU

I

Security Cla••ification

, 4. LINK A LINK.KEY wO"O.

LINK C

"OLE WT "OLII WT ..OLII WT

Liquid-Filled ProjectilesRotating LiquidsEigenfrequencies of Rotating LiquidsStability of Projectiles

J~

,j}- ..

I I

____________________________... ... ..._.1UNCLASSIFIED

8ecwtty ClaaalficadOil